Platonic solids generate their four-dimensional analogues

We show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan–Dieudonne theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, the $F_4$ root system and the 600-cell. In particular, these polytopes have ‘mysterious’ symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as $A_1\oplus I_2(n)$ which induces $I_2(n)\oplus I_2(n)$, explaining the existence of the grand antiprism and the snub 24-cell, as well as their automorphism symmetries. These results are discussed in the wider mathematical context of Arnold’s trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals, viruses and carbon onions, as well as four-dimensional geometries arising for instance in Grand Unified Theories and String and M-theory.